I did a quick search and it seems like tea tree oil contains lots of menthol-like terpenes (once a chemist, allways a chemist), and I'm already using menthol as it dries out the sores so it doesn't spread. So yeah, I guess tea tree oil would work as well.

I noe only have one (1) sore on my bottom lip. Hope it'll dissappear before a new one appears.